what is novometric data analysis?

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What is Novometric Data Analysis?

Paul R. Yarnold, Ph.D. Optimal Data Analysis, LLC

Novometric (Latin: new measure) statistical theory is introduced.

The mathematical modeling methodology which

is called optimal discriminant analysis or ODA

identifies nonparametric statistical models that

explicitly maximize the (weighted) accuracy of

predicted class assignments for the sample.1-5

It is crucial to understand that accuracy

is not the same thing as explained variation or

maximum likelihood. Neither of those centuries-

old objective functions motivated development

of methods which predict data very well, not

even if effect sizes approach their theoretically

maximum-possible values—thereby making p-

values incomputable due to sparsity issues.6-9

In contrast, what one seeks using ODA

is predictive precision. For example, imagine

each individual case (“observation”) in a sample

is either a member of class A or class B. Further

imagine a statistical model is obtained to predict

whether each individual case in the sample is a

member of class A or class B. ODA finds the

model which makes the most accurate possible

predictions into classes A vs. B for the sample,

given the data. The most accurate solution that

exists for the sample is the optimal solution.1-5

This paper briefly reviews etiology and

ontogenesis of novometric (Latin: new measure-

ment) statistical theory, the state-of-the-art ODA

methodology.5 Discussion first addresses crucial

aspects of ODA classification models and of

classification performance indices.

Classification Models and Performance

ODA is used in an effort to accurately predict a

class variable, known as a dependent variable in

legacy methods. Attributes, called independent

variables in legacy methods, are used to predict

class variables. Class variables and attributes

may be ordered or (multi)categorical10-12

; any

positive number may be used as an observation-

level weight (e.g., propensity score13-15

); models

may be static16,17

or dynamic18-26

; a priori hy-

potheses may be exploratory or confirmatory

(i.e., one- or “two”-tailed); exact p-values and

statistical power do not require distributional

assumptions, and are easily computed27-34

; and

optimal methods are rapidly and successfully

adopted by research faculty and students.35

Early use of maximum-accuracy pre-

dictive methods often addressed engineering

objectives. Prevalent then, and still today, the

tradition was to maximize the (weighted) pro-

portion of the total sample correctly classified,

without considering the class membership status

of individual observations.4,36-41

The associated

Percentage Accuracy in Classification or PAC

index ranges between 0% and 100% correct,

however this index is not normed vs. chance.

The focus of modern research, the Effect

Strength for Sensitivity or ESS index of classifi-

cation accuracy—a function of the model mean

sensitivity across classes—is adjusted by prior


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196

odds in order to eliminate the effect of chance.4

ESS=0 is the level of classification accuracy

that is expected by chance; ESS=100 is perfect

(errorless) classification; and -100< ESS<0

indicates classification accuracy worse than is

expected by chance.4,42-47

Regarding the theoretical meaning of

ESS, the findings of original Monte Carlo

investigations, in conjunction with advances

made in subsequent research, motivate the

continuing use of the categorical ordinal ESS

descriptors as originally proposed—which have

passed the test of time: ESS<25 is a relatively

weak effect; ESS<50 a moderate effect; ESS<75

is a relatively strong effect; ESS<90 is a strong

effect; and ESS>90 is a very strong effect.4

Models with a high ESS are favored in applied

translational and precision forecasting applica-

tions, owing to their adaptability to the diversity

reflected by legions of individual case profiles.

As ESS adjusts classification accuracy to

compensate for the role of chance, the distance

or D statistic adjusts ESS to compensate for the

role of model complexity—operationalized for-

mulaically in terms of number of sample strata

identified by the GO model.5 D indicates the

additional number of effects (which return the

mean ESS) needed to obtain a perfectly accurate

model.47-49

Models with a small D are favored in

theoretical applications, because they retain

only the empirically strongest findings.

Novometric Statistical Theory5,50,51

Novometric theory postulates the following set

of four axioms:

Axiom 1: For a random statistical sample S1

consisting of a class variable, one or more

attributes and a weight, corresponding exact

discrete 95% confidence intervals obtained for

model and for chance do not overlap (i.e., a

“statistically significant effect” exists).52-56

Axiom 2: In applications involving more than

one attribute, the subset of attributes in S1 which

yields the globally optimal (GO) model having

the lowest D statistic is identified via structural

decomposition analysis (SDA), which involves

iterative application of ODA in a manner that is

conceptually analogous to principal components

analysis, but that explicitly maximizes classifi-

cation accuracy—instead of variance.4,35,57-61

Note: In my experience using ODA

software, a primary utility of Axiom 2 (which

required four years to authenticate) lies in the

analysis of data sets having large N and many

attributes bearing a weak- to moderate-strength

relationship to the class variable, that degrades

in LOO analysis. Analyzing such data it is not

inconceivable to use CPU days in unsuccessful

attempts (attributable to researcher fatigue) to

identify the initial CTA model. In my personal

experience, if I can obtain a descendant family

(see Axiom 3) in reasonable time-on-task (an

hour or less), then I often forego Axiom 2.

Axiom 3: The GO model for S1 is found in the

descendant family of models that is obtained by

applying the minimum denominator search

algorithm (MDSA) to an initially-unrestricted

enumerated classification tree analysis (CTA)

model configured to predict the class variable

using only the attribute(s) selected by SDA. The

MDSA algorithm enables discovery of all CTA

models in S1 which vary as a function of com-

plexity and that originate from an unadulterated

(initially unrestricted) model.5,13,18

Note: I learned to not make assumptions

when obtaining a descendant family. Suffice it

to say, I continue to augment the minimum de-

nominator until the software returns a message

stating “no model found”.

Axiom 4: Validity is assessed by using the GO

model to classify an independent random S2 (if

possible), and/or for S1 by cross-generalizability

methods such as hold-out, leave-one-out (LOO)

one-sample jackknife, K-of-N jackknife, split-

half, multi-sample and/or bootstrap methods for


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static data4, and using test-retest

62-67, survival

(time-to-event)21,23

, little jiffy68,69

, and weighted

Markov70-74

methods with dynamic (repeated-

measures) data. These types of analyses enable

researchers to identify the limits of cross-gener-

alizability of GO models when they are used to

classify independent random samples.4,5

Discussion now considers two simple

real-world examples of novometric analysis.

Gender and Cancer Incidence

Imagine evaluating if the cancer incidence rate

(CIR) for all cancer types combined, assessed

across numerous independent sites, differs by

gender—male vs. female.50

If using a legacy methodology such as

t-test75-77

or logistic regression9,78-80

to compare

CIR between gender, one’s conclusion would

necessarily be: (a) the absence of a statistically

significant effect indicates the CIR of males and

females does not differ; or (b) the presence of a

statistically significant effect indicates males

have a (higher or lower) CIR vs. females.

In contrast, having only two options as a

possible analytic solution in this example is not

a restriction for novometric analysis.

Data were drawn from the Surveillance,

Epidemiology and End Results (SEER) Pro-

gram, which collects and publishes cancer inci-

dence and survival data to assemble and report

estimates of cancer incidence, survival, mortal-

ity, other measures of the cancer burden, and

patterns of care, in the USA. SEER statistics

routinely include information specific to race/

ethnic populations and other populations de-

fined by age, gender and geography. Herein,

CIR is the number of new cancers of a specific

site that occurs in a specified population in one

year, expressed as the number of new cancers

for every 100,000 population at risk.

Analysis involved N=608 observations.

Assuming proportional reduction in sample size

across successive parses: a single binary parse

creates two strata each having N=304 observa-

tions; two binary parses create four strata each

having N=152 observations; and three binary

parses create eight strata each having N=76

observations. This sample size is more than

sufficient34

to achieve greater than 90% power

to identify a moderate effect with p<0.05.

Table 1 summarizes findings of novo-

metric analysis using gender to parse CIR data.

The initial (6 strata) CTA model that emerged

had moderate ESS: its point estimate (33.2) and

the upper (41.2) and lower (25.4) bounds of the

95% CI for model-based ESS all indicate mod-

erate effect. The 95% CIs for chance-based ESS

lie substantially beneath corresponding 95% CIs

for model-based ESS. Note that since no two-

strata model comparing all males vs. all females

exists in the descendant family, this is a statisti-

cally unmotivated, exploratory comparison.

Table 1: Parsing Cancer Incidence by Gender:

All Sites Combined

Strata MinD ESS Efficiency D

-----------------------------------------------------------------

6 2 33.2 5.54 12.1

25.4-41.2 4.22-6.87 8.6-17.6

0.33-7.57 0.06-1.26 73.3-1812

5 63 33.2 6.64 10.1

25.3-41.2 5.05-8.23 7.1-14.8

0.33-6.91 0.07-1.38 67.4-1510

3 80 31.9 10.6 6.4

22.8-40.6 7.60-13.5 4.4-10.2

0.33-7.57 0.11-2.52 36.6-906

-----------------------------------------------------------------

Note: Strata is number of CTA model endpoints.

MinD (for minimum denominator) is the smallest

sample size for any strata. Efficiency, defined as

ESS/Strata, is a normed index of relative strength

of the class variable(s) used in identifying sample

strata. Results for every step of MDSA analysis

are tabled. For each model the first line is a point

estimate; the second line gives a 95% confidence

interval (CI) for the discrete distribution obtained

for the model by bootstrap analysis; and the third

line is a 95% CI for chance obtained using Monte

Carlo analysis with 100,000 iterations. Values for

model efficiency were computed by dividing ESS

by the number of strata.


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As seen in Table 1 the initial CTA model

identified six strata, the smallest of which repre-

sented only two observations. The 95% CI for

this endpoint (not tabled) included 0%, render-

ing the CTA model redundant.

In step two of the minimum denominator

selection algorithm, a LOO-stable five-strata

CTA model was obtained yielding moderate

ESS equivalent to that of the initial six-strata

model, as assessed by point estimate and by

95% CI overlap. The five-strata model had

6.64/5.54 or 19.9% greater efficiency as

assessed by point estimate, however this

difference wasn’t statistically significant

because the five- and six-strata 95% CIs for

efficiency overlapped. The smallest strata for

this model had N=63 patients.

In step three the final LOO-stable model

in the descendant family identified three patient

strata. Based on point estimates the final model

achieved 31.9/33.2 or 96% of overall accuracy

(ESS) vs. the other models. And, examining

95% CIs for model-based ESS reveals that all

models had comparable ESS (the three-strata

model had greater efficiency vs. the six-strata

model). However, because the lower-bound of

ESS of the three-attribute model was 22.8 (or

relatively weak), the ESS for the three-strata

model indicates a weak-moderate effect.

Having achieved comparable strength,

and greater parsimony and efficiency vs. other

models in the descendant family, the three-strata

model is selected as being the GO model of the

relationship between gender and all-site cancer

incidence. Illustrated in Figure 1, circles (model

nodes) represent attributes (cancer incidence);

arrows indicate model branches; rectangles are

model end-points and represent sample strata;

numbers adjacent to arrows are the value of the

cutpoint for the node; numbers beneath nodes

are the exact per-comparison Type I error rate

for the parse; the number of observations classi-

fied in each endpoint is indicated beneath the

endpoint; and the percentage of male observa-

tions is given inside the endpoint.

Figure 1: Three-Strata CTA Model Predicting

Cancer Incidence by Gender, All Cancer Sites

Cancer

Incidence

53.45%

Males

30.83%

Males

98.75%

Males

N=275 N=253 N=80

p<0.0001 p<0.0001

<249.439

<2066.0435

>2066.0435

As seen, the sample strata having lowest

cancer incidence (<0.25%) was approximately

equally represented by males and females, and it

comprised 275/608 or 45% of the total sample.

In contrast, the sample strata with highest cancer

incidence (>2.06%) was dominated (98.8%) by

males, and comprised 13% of the sample. And,

the sample strata having an intermediate cancer

incidence (0.26%-2.065%) was 69.2% female

and comprised 42% of the total sample.

Fear, Specific Recommendation, and

Inoculation Shot-Taking Behavior

Classic research tested the a priori hypothesis

that fear and specificity of recommendation

synergistically influence a person’s decision to

have a tetanus inoculation.81

Data analysis by 2

missed the hypothesized interaction: “…specific

plans for action influence behavior while level

of fear does not” (p. 27).

For these data, CTA with shot taking

(yes vs. no) used as class variable, and fear and

specific recommendation as attributes, found a

relatively strong, cross-generalizable two-strata

model (Figure 2) supporting the a priori hypoth-

esis that fear and specific recommendations syn-

ergistically predict shot-taking behavior.82

The

confusion matrix for this three-strata model is

presented in Table 2.


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199

Figure 2: Fear and Specific Recommendation

Predict Shot-Taking Behavior

Fear

Specific

Recommendation

0%

Took Shot

3.3%

Took Shot

27.6%

Took Shot

p<0.001

p<0.014

N=30 N=29

N=90

No Yes

No Yes

Table 2: Confusion Matrix for CTA Model,

Factorial Design, Fear and Specific

Recommendation

Predicted Behavior

No Shot Shot

Actual No Shot 119 21 85.0%

Behavior Shot 1 8 88.9%

A statistically reliable, reproducible, and

strong effect (ESS=73.9) emerged: as seen, 7 in

8 of the people refusing the shot, and 8 in 9 of

the people accepting the shot, were correctly

predicted by the model. Consistent with the

original hypothesis, fear and specific recom-

mendation interact to affect shot-taking behav-

ior. As the root variable, fear plays the dominant

role. When there is no fear, 0% of 90 people

took the shot. When there is fear, but no specific

recommendation is given, 3.3% of 30 people

took the shot. However when there is fear and

specific recommendation, 27.6% of 29 people

took the shot.

Coda

In the event that this brief introduction leaves

one wanting to know more, I recommend a few

papers discussing modeling ordered class

variables83-85

, matrix display of findings86,87

, and

generating novometric confidence intervals

using R software.88

In addition, articles are

available which contrast novometric analysis

with legacy methods such as ANOVA89-93

,

Friedman test94

, log-linear analysis95-99

, logit

analysis100

, logistic regression101

, Markov

analysis102,103

, regression/correlation104-109

,

polychoric correlation110,111

, Probit analysis112

,

recursive causal analysis113

, sign test114

,

stepwise analysis115

, temporal analysis116-119

,

and Yule’s Q.120

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76Yarnold PR (2015). UniODA vs. t-Test: Low

brain uptake of L-[11

C]5-hydroxytryptophan in

major depression. Optimal Data Analysis, 4,

146-147.

77Yarnold PR (2019). ODA vs. t-test: Lysozyme

levels in the gastric juice of patients with peptic

ulcer vs. normal controls. Optimal Data

Analysis, 8, 112-113.


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204

78Yarnold PR (2014). UniODA vs. logistic

regression analysis: Serum cholesterol and

coronary heart disease and mortality among

middle aged diabetic men. Optimal Data

Analysis, 3, 17-18.

79Yarnold PR (2015). UniODA vs. logistic

regression and Fisher’s linear discriminant

analysis: Modeling 10-year population change.


80Linden A, Bryant FB, Yarnold PR (2019).

Logistic discriminant analysis and structural

equation modeling both identify effects in

random data. Optimal Data Analysis, 8, 97-102.

81Leventhal H, Singer R, Jones S (1965). Effects

of fear and specificity of recommendation upon

attitudes and behavior. Journal of Personality

and Social Psychology, 2, 20-29.

82Yarnold PR (2016). CTA vs. not chi-square:

Fear and specific recommendations do

synergistically affect behavior. Optimal Data

Analysis, 5, 108-111.

83Yarnold PR Linden A (2016). Novometric

analysis with ordered class variables: The

optimal alternative to linear regression analysis,


84Yarnold PR Bennett CL (2016). Novometrics

vs. correlation: Age and clinical measures of

PCP survivors, Optimal Data Analysis, 5, 74-

78.

85Yarnold PR Bennett CL (2016). Novometrics

vs. multiple regression analysis: Age and

clinical measures of PCP survivors, Optimal


86Yarnold PR (2016). Matrix display of pairwise

novometric associations for ordered variables.


87Yarnold PR Batra M (2016). Matrix display of

pairwise novometric associations for mixed-

metric variables. Optimal Data Analysis, 5, 104-

107.

88Rhodes JN, Yarnold PR (2020). Generating

novometric confidence intervals in R: Bootstrap

analyses to compare model and chance ESS.


89Yarnold PR (2016). Novometrics vs. ODA vs.

One-Way ANOVA: Evaluating comparative

effectiveness of sales training programs, and the

importance of conducting LOO with small

samples. Optimal Data Analysis, 5, 131-132.

90Yarnold PR (2016). Novometric analysis vs.

MANOVA: MMPI codetype, gender, setting,

and the MacAndrew Alcoholism scale. Optimal


91Yarnold PR (2018). ANOVA with one

between-groups factor vs. novometric analysis.


92Yarnold PR (2018). ANOVA with two

between-groups factors vs. novometric analysis.


93Yarnold PR (2018). ANOVA with three

between-groups factors vs. novometric analysis.


94Yarnold PR (2018). Friedman test vs. ODA vs.

novometry: Rating violin excellence. Optimal


95Yarnold PR (2016). Novometric analysis

predicting voter turnout: Race, education, and

organizational membership status. Optimal Data

Analysis, 5, 194-197.


GenODA vs. log-linear model: Temporal

stability of the association of presidential vote

choice and party identification. Optimal Data

Analysis, 5, 204-207.


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205


ODA vs. log-linear model in analysis of a two-

wave panel design: Assessing temporal stability

of Catholic party identification in the 1956-1960

SRC panels. Optimal Data Analysis, 5, 208-212.

98Yarnold PR (2016). Novometric vs. log-linear

model: Intergenerational occupational mobility

of white American men. Optimal Data

Analysis, 5, 218-222.

99Yarnold PR (2016). Novometric vs. log-linear

analysis: Church attendance, age and religion.


100Yarnold PR (2016). Novometric vs. logit

analysis: Abortion attitude by religion and time.


101Yarnold PR (2016). Novometric models of

smoking habits of male and female friends of

American college undergraduates: Gender,

smoking, and ethnicity. Optimal Data

Analysis, 5, 146-150.

102Yarnold PR (2017). Novometric analysis of

transition matrices to ascertain Markovian order.


103Yarnold PR (2017). Novometric comparison

of Markov transition matrices for heterogeneous

populations. Optimal Data Analysis, 6, 9-12.

104Yarnold PR (2016). Novometrics vs.

regression analysis: Literacy, and age and

income, of ambulatory geriatric patients.



regression analysis: Modeling patient

satisfaction in the Emergency Room. Optimal


106Linden A, Yarnold PR (2018). Comparative

accuracy of a diagnostic index modeled using

(optimized) regression vs. novometrics. Optimal


107Yarnold PR (2019). Multiple regression vs.

novometric analysis of a contingency table.


108Yarnold PR (2019). Regression vs.

novometric analysis predicting income based on

education. Optimal Data Analysis, 8, 81-83.

109Yarnold PR (2019). Regression vs.

novometric-based assessment of inter-examiner

reliability. Optimal Data Analysis, 8, 107-111.

110Yarnold PR (2016). Novometric vs. ODA

reliability analysis vs. polychoric correlation

with relaxed distributional assumptions: Inter-

rater reliability of independent ratings of plant

health. Optimal Data Analysis, 5, 179-183.


polychoric correlation: Number of lambs born

over two years. Optimal Data Analysis, 5, 184-

185.

112Yarnold PR (2016). Novometric vs. logit vs.

probit analysis: Using gender and race to predict

if adolescents ever had sexual intercourse.


113Yarnold PR (2016). Novometric vs. recursive

causal analysis: The effect of age, education,

and region on support of civil liberties. Optimal


114Yarnold PR (2016). Using novometrics to

disentangle complete sets of sign-test-based

multiple-comparison findings. Optimal Data

Analysis, 5, 175-176.

115Yarnold PR, Linden A (2019). Novometric

stepwise CTA analysis discriminating three

class categories using two ordered attributes.


116Yarnold PR (2017). Novometric comparison

of patient satisfaction with nurse responsiveness

over successive quarters. Optimal Data

Analysis, 6, 13-17.


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206

117Yarnold PR (2017). Novometric pairwise

comparisons in consolidated temporal series.


118Linden A, Yarnold PR (2018). The Australian

Gun Buyback Program and rate of suicide by

Firearm. Optimal Data Analysis, 7, 28-35.

119Yarnold PR (2020). Modeling an individual’s

weekly change in RG-score via novometric

single-case analysis. Optimal Data Analysis, 9,

114-121.

120Yarnold PR (2016). Novometrics vs. Yule’s

Q: Voter turnout and organizational

membership. Optimal Data Analysis, 5, 198-

199.

Author Notes

No conflicts of interest were reported.

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